[Paper Review] The boundary analog of the Carathéodory-Schur interpolation problem

V. Bolotnikov|arXiv (Cornell University)|Aug 19, 2010

Holomorphic and Operator Theory13 references6 citations

TL;DR

This paper establishes a boundary analog of the classical Carathéodory-Schur interpolation problem for Schur-class functions, providing necessary and sufficient conditions for the existence of a Schur function with prescribed nontangential boundary expansion at a point on the unit circle. The key result characterizes solvability via a positive semidefinite matrix constructed from the boundary data and two auxiliary parameters, extending classical interior interpolation to the boundary setting with explicit uniqueness and finiteness conditions.

ABSTRACT

Characterization of Schur-class functions (analytic and bounded by one in modulus on the open unit disk) in terms of their Taylor coefficients at the origin is due to I. Schur. We present a boundary analog of this result: necessary and sufficient conditions are given for the existence of a Schur-class function with the prescribed nontangential boundary expansion $f(z)=s_0+s_1(z-t_0)+\ldots+s_N(z-t_0)^N+o(|z-t_0|^N)$ at a given point $t_0$ on the unit circle.

Motivation & Objective

• To extend the classical Carathéodory-Schur interpolation problem from interior points in the unit disk to boundary points on the unit circle.
• To characterize the existence of Schur-class functions with a given nontangential boundary expansion of order $ N $ at a point $ t_0 \in \mathbb{T} $.
• To determine conditions under which such a function exists, and when it is unique or part of an infinite family.
• To identify the unique solution as a finite Blaschke product in the case of uniqueness, and to describe the structure of the solution set in the indeterminate case.

Proposed method

• The problem is formulated as finding a Schur-class function $ f \in \mathcal{S} $ with prescribed nontangential boundary limits $ f_j(t_0) = s_j $ for $ j = 0, \dots, N $, equivalent to the asymptotic expansion $ f(z) = \sum_{j=0}^N s_j (z - t_0)^j + o(|z - t_0|^N) $ as $ z \to t_0 $ nontangentially.
• A key construction involves a positive semidefinite matrix $ \mathbb{P}^\mathbf{s}_n $ derived from the coefficients $ s_0, \dots, s_N $, and an auxiliary matrix involving higher-order derivatives and a parameter $ R_0 $ related to the boundary behavior.
• The existence of a solution is determined by the positivity of a structured matrix $ \mathbb{P}^\mathbf{s}_n $, with additional conditions involving the complex number $ R_0 $ and the difference $ p^{\mathbf{s}}_{n+1,n} - \overline{p}^{\mathbf{s}}_{n,n+1} $.
• The analysis uses the Carathéodory-Julia theory and properties of Schur functions, particularly the behavior of $ \frac{1 - |f(z)|^2}{1 - |z|^2} $ near the boundary.
• The proof relies on a recursive construction of extensions of the interpolation data and the use of a function $ \mathcal{E} \in \mathcal{S} $ satisfying $ \mathcal{E}(t_0) = R_0 $, with solvability depending on whether $ |R_0| < 1 $.
• The uniqueness and finiteness of solutions are tied to the rank of the matrix $ \mathbb{P}^\mathbf{s}_n $, with a unique solution being a finite Blaschke product when $ \mathbb{P}^\mathbf{s}_n $ is singular.

Experimental results

Research questions

• RQ1 Under what conditions does a Schur-class function exist with a prescribed nontangential boundary expansion of order $ N $ at a boundary point $ t_0 \in \mathbb{T} $?
• RQ2 When is such a function unique, and when are there infinitely many solutions?
• RQ3 How does the structure of the solution set (finite or infinite) relate to the positivity and singularity of an associated matrix constructed from the boundary data?
• RQ4 What role do the auxiliary parameters $ R_0 $ and $ c_n(t_0) $ play in determining solvability?
• RQ5 How does the boundary interpolation problem relate to the classical interior Carathéodory-Schur problem, and what are the key differences in the solution criteria?

Key findings

• A Schur-class function with the prescribed boundary expansion at $ t_0 \in \mathbb{T} $ exists if and only if a certain matrix $ \mathbb{P}^\mathbf{s}_n $ constructed from the coefficients $ s_0, \dots, s_N $ is positive semidefinite, and an auxiliary condition involving $ R_0 $ and the difference $ p^{\mathbf{s}}_{n+1,n} - \overline{p}^{\mathbf{s}}_{n,n+1} $ is satisfied.
• If $ |s_0| < 1 $, there are infinitely many solutions, consistent with the classical case.
• If $ |s_0| = 1 $ and $ \mathbb{P}^\mathbf{s}_n $ is singular, then there exists a unique solution, which is a finite Blaschke product of degree equal to the rank of $ \mathbb{P}^\mathbf{s}_n $.
• When $ |s_0| = 1 $, $ \mathbb{P}^\mathbf{s}_n > 0 $, and $ N = 2n - 1 $, there are infinitely many solutions.
• If $ |s_0| = 1 $, $ \mathbb{P}^\mathbf{s}_n > 0 $, and $ N > 2n $, then solutions exist only if $ t_0(p^{\mathbf{s}}_{n+1,n} - \overline{p}^{\mathbf{s}}_{n,n+1}) > 0 $, and in this case there are infinitely many solutions; otherwise, no solution exists.
• In the case where $ |s_0| = 1 $, $ \mathbb{P}^\mathbf{s}_n > 0 $, and $ N = 2n $, solutions exist if and only if $ p^{\mathbf{s}}_{n+1,n} = \overline{p}^{\mathbf{s}}_{n,n+1} $, and in this case there are infinitely many solutions.